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\@writefile{toc}{\contentsline {section}{\tocsection {}{1}{More Strange Result}}{1}}
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\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces Surprise: there is a limit to how close we can localize the wave in both x and p-space.}}{1}}
\@writefile{toc}{\contentsline {section}{\tocsection {}{2}{Multiple Particles: Making the transition from 1 particle systems to N particle systems}}{1}}
\@writefile{toc}{\contentsline {paragraph}{\tocparagraph {}{}{So far we've only considered the Schr\"odinger equation for 1 particle. While we can look at an N particle case, for our needs right now, it will suffice to look at the formalism for a 2 particle Schr\"odinger equation. So we are going from $\psi (\mathaccentV {vec}17E{x},t)\to \psi (\mathaccentV {vec}17E{x_{(1)}},\mathaccentV {vec}17E{x_{(2)}},t)$, while considering the three dimensional case.}}{1}}
\@writefile{toc}{\contentsline {paragraph}{\tocparagraph {}{}{There is a strange feature that we should now consider. In the one particle case of $\psi (\mathaccentV {vec}17E{x},t)$, at each point in 3-space, this function gives us a number. This function $\psi $ is a map $\psi : \mathbb  {R}^3 \to \mathbb  {C}$. In the 2 particle case, we will need 6 pieces of data, and our wave function here is a map $\psi _2:\mathbb  {R}^6 \to \mathbb  {C}$. By now we should see that in the N-particle case, we'd have a wave function $\psi _N: \mathbb  {R}^{3N}\to \mathbb  {C}$, given by $\psi (\mathaccentV {vec}17E{x_{(1)}},...,\mathaccentV {vec}17E{x_{(N)}},t)$. Think about this for a second. This is astounding! If there were $10^{86}$ particles in the universe, and we assume that there is only 3-space, then the wave function describing all particles would be a map from $R^{3\times 10^{86}}\to \mathbb  {C}$.}}{1}}
\@writefile{toc}{\contentsline {paragraph}{\tocparagraph {}{}{To further address the strange features of the wave function, we will need to move onto Fourier Analysis.}}{1}}
\@writefile{toc}{\contentsline {section}{\tocsection {}{3}{The Pure Mathematical Facts of Fourier Analysis}}{1}}
\@writefile{toc}{\contentsline {paragraph}{\tocparagraph {}{}{The basic idea to begin with begins with the linearity of Schr\"odinger's equation. Because of linearity, we can combine solutions to get new solutions.}}{1}}
\@writefile{toc}{\contentsline {paragraph}{\tocparagraph {}{}{The punchline to all of this? Ignoring the physical interpretation for a moment, if we're given any sufficiently well behaved function $\psi (\mathaccentV {vec}17E{x},t)$ (that is, $\DOTSI \intop \ilimits@ _\mathbb  {R}|\psi (\mathaccentV {vec}17E{x},t)|^2 <\infty $), then there exists another function, $\phi (k)$ s.t.}}{2}}
\@writefile{toc}{\contentsline {paragraph}{\tocparagraph {}{}{To build up to this result, let us start with Fourier Series}}{2}}
\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{3.1}{Fourier Series}}{2}}
\@writefile{toc}{\contentsline {paragraph}{\tocparagraph {}{}{To begin, consider any periodic function, s.t $\psi (x)=\psi (x+\iota )$. Since sine and cosine are also periodic, it would be a natural guess that any such $\psi $ could be expressed as a sum of sine and cosine. That is:}}{2}}
\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{3.2}{To Generalize}}{2}}
\@writefile{toc}{\contentsline {paragraph}{\tocparagraph {}{}{We want to relax that condition that $\psi $ is periodic. To do so, start with $\psi (x)=\psi (x+\iota )$, and then let $\iota \to \infty $. What do we see?}}{2}}
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\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{3.3}{Summary}}{3}}
\@writefile{toc}{\contentsline {paragraph}{\tocparagraph {}{}{Any $\psi (x)$ can be written as a linear combination over sines and cosines, since $e^{ikx}= \qopname  \relax o{cos}(kx)+i\qopname  \relax o{sin}(kx)$.}}{3}}
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\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces When $\rho $ is large, $\psi $ is sharply peaked, and when $\rho $ is small $\psi $ is more spread out. }}{3}}
\@writefile{toc}{\contentsline {section}{\tocsection {}{4}{The Physical Interpretation}}{3}}
\@writefile{toc}{\contentsline {paragraph}{\tocparagraph {}{}{With $\psi _{\rho }(x)$ as a wave function for a particle, the particle has a larger probability to be located at or near origin x=0. As $\rho \to $ large, the particle is more tightly localized near x=0. As $\rho \to $small, the particle becomes more spread out. }}{3}}
\@writefile{toc}{\contentsline {paragraph}{\tocparagraph {}{}{But what is the meaning of $\phi (k)$? We still require $\DOTSI \intop \ilimits@ _\mathbb  {R}|\phi (k)|^2\tmspace  +\thinmuskip {.1667em}dk=1$, since this is a probability wave. However, $\phi (k)$ is the probability that the particle has momentum $P=\hbar k$.}}{3}}
\@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces Here $\rho $ is large, and so our $\psi $ is spiked, while $\phi $ is spread out.}}{3}}
\@writefile{lof}{\contentsline {figure}{\numberline {4}{\ignorespaces Here $\rho $ is small, and so $\psi $ is spread out, while $\phi $ is spiked. This is another encounter with \emph  { the uncertainty principle}}}{3}}
